Atlas › Modular forms › Weight 2 › Level 694 › 694.2.a › 694.2.a.a

Newform orbit 694.2.a.a

Trace form

The 5 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$5q + 5q^{2} - 5q^{3} + 5q^{4} - 7q^{5} - 5q^{6} - 9q^{7} + 5q^{8} + 2q^{9} - 7q^{10} - 11q^{11} - 5q^{12} + 2q^{13} - 9q^{14} - 3q^{15} + 5q^{16} - 14q^{17} + 2q^{18} - 15q^{19} - 7q^{20} + 2q^{21} - 11q^{22} - 6q^{23} - 5q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 5. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + T_{5}$ , whose minimal polynomial is

$x^{5} + 12x^{4} + 51x^{3} + 88x^{2} + 48x + 1$

Atkin–Lehner signs

$p$	$w_p$
2	$-1$
347	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = +1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = -1$ : each L-function in the orbit has odd analytic rank.

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	5	−5	−7	−9	−11	2	−14	−15	−6	−7	−14	7	−19	7	−9	11	−4	13	−3	−13	11	0	2	−12	−24

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.222 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 347, 694]: [0, 1, 30, 89] (certified at ell = 2147481179)
T = [(3, 1), (5, 1)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 3 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(694, 2)[0]   # 694.2.a.a

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 694:2 1000